Definition: A valid argument is a deductive argument that succeeds in providing decisive logical support.

A valid argument is thus a deductive argument – an argument that attempts to establish conclusive support for its conclusion – that succeeds.

Definition: An invalid argument is a deductive argument that fails in providing conclusive support.

For deductive arguments, you answer “yes” to the question “Do the premises provide enough logical support for the conclusion?” if the argument is valid, and you answer “no” if otherwise.

Take the following deductive argument:

Patrick’s jeans are blue, therefore, Patrick’s jeans are coloured.

Is it possible for the premises to be true and the conclusion to be false? If my jeans are blue, then they have a colour. If they have a colour, then they’re coloured. Hence, it is impossible for the premise (Patrick’s jeans are blue) to be true, and the conclusion (Patrick’s jeans are coloured) to be false. Therefore, the argument is valid.

How about this one?

If you throw a dice, either it lands on six or it doesn’t. So the dice has a 50% chance of landing on six.

Some people believe that, but this is an invalid argument. What is the probability for a dice to land on six? There are six faces and the dice is likely to land on any of them. Since six only shows on one face, there’s only a one out of six chance that the dice will land on six. And one out of six is a lot less than 50%. It is thus possible for the premise of the argument to be true, but the conclusion false.

Arguments can be valid even if they are rubbish:

If there is a purple elephant in the hall, then I am a giant turkey. There is a purple elephant in the hall. Therefore, I’m a giant turkey.

This argument is nonsensical, but it’s valid. If the premises were true, the conclusion would be guaranteed to be true. You need to be careful here. ‘Valid’ does not necessarily mean good or bad. It just means succeeding in establishing conclusive support for its conclusion. Of course, the premises of this argument are false. But claiming that an argument is valid is not to claim that the premises are true. Validity is about succeeding in providing conclusive support for the conclusion, if the premises were true.

For non-deductive arguments, we don’t talk about valid and invalid arguments, we talk instead about strong and weak arguments.

Definition: A strong argument is a non-deductive argument that succeeds in providing probable, but not conclusive, logical support for its conclusion. A weak argument is a non-deductive argument that fails to provide probable support for its conclusion.

If an argument is weak, you’d be better off throwing a coin to know if the conclusion is true, and that’s far from succeeding in providing reasons for a conclusion. So if the conclusion is unlikely to be true when the premises are true, then the argument is weak. Game over.

But how are we to decide when an argument is strong and not weak?

The answer to this question is contextual. As a lecturer, my standards are very strict. I’m extremely pedantic, and I’m going to point out every mistake you made. My goal is to make sure that you learn from your mistakes. But if I do that when I’m at a party with my friends, I would not be very popular, now would I? I need to change my standards there. When I’m in the court of law, I need to have strong standards, beyond reasonable doubt. So establishing that an argument is strong in court is quite demanding. We want to minimise the mistakes we make.

Let’s try with some examples. Strong or weak?

97% of vegetarians are healthy. Madison is a vegetarian. Therefore, Madison is probably healthy.

If the premises are true, what are the chances that Madison is healthy? 97%! She might not be. She might be amongst the 3% that are not healthy, but it’s quite unlikely. So if the premises are true, it’s very likely that the conclusion is also true—but it may be false. Hence, this argument is strong.

Bailey’s father is a plumber, so Bailey’s father has a van.

I think this one is also strong. Here’s my reasoning. Being a plumber is the kind of job that requires you to go around with tools, and tools that are messy and get dirty and everything, and you don’t want to put it in a nice little car; you need space. In my experience, plumbers typically have vans. So then it would be quite reasonable to expect Bailey’s father to have a van if he’s a plumber.